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Convex Chance-Constrained Stochastic Control under Uncertain Specifications with Application to Learning-Based Hybrid Powertrain Control

Teruki Kato, Ryotaro Shima, Kenji Kashima

TL;DR

This work tackles stochastic control when control specifications themselves are uncertain, formulating a strictly convex chance-constrained optimization that jointly optimizes inputs and risk allocation to guarantee probabilistic constraint satisfaction even under non-Gaussian uncertainties. By relaxing joint chance constraints with Boole's inequality and introducing a convex risk-allocation regularizer, the authors prove existence, convexity, and (under mild conditions) uniqueness of solutions, and extend the framework to learning-based exactly linearizable nonlinear models. The approach is applied to model predictive control of a hybrid powertrain, demonstrating reduced conservatism and reliable constraint satisfaction despite uncertainty in future requested speed. The results indicate the method's potential for robust, data-driven control of complex systems where specifications are uncertain and non-Gaussian disturbances are present, with practical impact on energy efficiency and emissions in automotive powertrains.

Abstract

This paper presents a strictly convex chance-constrained stochastic control framework that accounts for uncertainty in control specifications such as reference trajectories and operational constraints. By jointly optimizing control inputs and risk allocation under general (possibly non-Gaussian) uncertainties, the proposed method guarantees probabilistic constraint satisfaction while ensuring strict convexity, leading to uniqueness and continuity of the optimal solution. The formulation is further extended to nonlinear model-based control using exactly linearizable models identified through machine learning. The effectiveness of the proposed approach is demonstrated through model predictive control applied to a hybrid powertrain system.

Convex Chance-Constrained Stochastic Control under Uncertain Specifications with Application to Learning-Based Hybrid Powertrain Control

TL;DR

This work tackles stochastic control when control specifications themselves are uncertain, formulating a strictly convex chance-constrained optimization that jointly optimizes inputs and risk allocation to guarantee probabilistic constraint satisfaction even under non-Gaussian uncertainties. By relaxing joint chance constraints with Boole's inequality and introducing a convex risk-allocation regularizer, the authors prove existence, convexity, and (under mild conditions) uniqueness of solutions, and extend the framework to learning-based exactly linearizable nonlinear models. The approach is applied to model predictive control of a hybrid powertrain, demonstrating reduced conservatism and reliable constraint satisfaction despite uncertainty in future requested speed. The results indicate the method's potential for robust, data-driven control of complex systems where specifications are uncertain and non-Gaussian disturbances are present, with practical impact on energy efficiency and emissions in automotive powertrains.

Abstract

This paper presents a strictly convex chance-constrained stochastic control framework that accounts for uncertainty in control specifications such as reference trajectories and operational constraints. By jointly optimizing control inputs and risk allocation under general (possibly non-Gaussian) uncertainties, the proposed method guarantees probabilistic constraint satisfaction while ensuring strict convexity, leading to uniqueness and continuity of the optimal solution. The formulation is further extended to nonlinear model-based control using exactly linearizable models identified through machine learning. The effectiveness of the proposed approach is demonstrated through model predictive control applied to a hybrid powertrain system.
Paper Structure (17 sections, 10 theorems, 37 equations, 5 figures, 2 tables)

This paper contains 17 sections, 10 theorems, 37 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: Deterministic control for a linear system
  3. Stochastic control accounting for uncertainty in control specifications for a linear system
  4. Extension to learning-based exactly linearizable models
  5. Application to hybrid powertrain control
  6. Numerical example
  7. Implementation of control accounting for uncertainty in requested vehicle speed
  8. Control methods to be compared
  9. Control results
  10. Risk-allocation behavior and evaluation of empirical violation probabilities
  11. Conclusion
  12. Probability inequalities
  13. Boole's inequality
  14. Cantelli's inequality
  15. Hoeffding's inequality
  16. ...and 2 more sections

Key Result

Theorem 1

For the chance-constrained stochastic control problem eq:joint_chance_obj--eq:joint_chance_const, which is a probabilistic counterpart of the deterministic optimal control problem eq:opt_control_lin_obj--eq:opt_control_lin_const, the following statements hold under Assumption ass:theta_distribution.

Figures (5)

  • Figure 1: Exactly linearizable model
  • Figure 2: Comparison of control results for the hybrid powertrain system: stochastic control with simultaneous risk optimization vs. deterministic control
  • Figure 3: Comparison of control results for the hybrid powertrain system: stochastic control with uniform risk allocation vs. deterministic control
  • Figure 4: Time evolution of risk allocation by constraint type in the proposed method, total allocated risk, and empirical violation probabilities (also shown: total allowable risk $\overline{\delta}$ and uniform allocation $\overline{\delta}/3$)
  • Figure 5: Time evolution of empirical violation probabilities by constraint type in deterministic control (the total allowable risk $\overline{\delta}$ is shown by a dashed line)

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1: Boole's inequality
  • Corollary 1
  • ...and 11 more